Tukey’s HSD for ANCOVA

Basic Approach

Following a significant ANCOVA, we can use the usual approach to Tukey’s HSD and the Tukey-Kramer follow up test, as described in Unplanned Comparisons, except that we need to modify the value of MS_W as follows:

where SS_B(x) and SS_W(x) are the sums of squares between and within the covariance groups (based on one-way Anova on the covariate data), i.e.

The standard error can then be calculated as

if all the group sizes are equal (to m). This is the Tukey HSD test. If the group sizes are unequal, the standard error is (i.e. the Tukey-Kramer test)

Example

Example 1: Repeat Example 1 of Contrasts for ANCOVA using the Tukey-Kramer test.

The results are shown in Figure 1.

Figure 1 – Tukey-Kramer for ANCOVA

Note that range M4:N6 references cells in Figure 1 of ANOVA Approach to ANCOVA, although the formula for MS_W in cell Q9 references Figure 3 of ANOVA Approach to ANCOVA.

Data Analysis Tool

See ANCOVA Analysis Tools for information about how to conduct Tukey’s HSD or Tukey-Kramer using the Real Statistics ANCOVA data analysis tool.

Reference

Howell, D. C. (2010) Statistical methods for psychology (7^th ed.). Wadsworth, Cengage Learning.
https://labs.la.utexas.edu/gilden/files/2016/05/Statistics-Text.pdf